This the info I received from a US Cork supplier. They buy their Cork directly from Portugal.
Can you Stain Cork?
Thank you Bob Balcombe

Hello Bob,

Yes, cork can be stained using a waterbased wood stain.

How deep of a penetration will you get? Lets say I am dyeing cork rings for fishing rod grips
Thanks

The water based stain will penetrate about 2mm depending on the density of the cork.
Thanks,
Fay.

This should answer some questions on staining Cork rings

Good Wraps Bob

Thanks Bob; I've been wondering about this for a while and never got around to testing it for myself.

Walt Davenport
"Roadhouse Rods"

With the stain only penetrating just the surface. I would think you still have to put a protective covering over the stain or it will ware off.

Good Wraps Bob

I've never tried to stain solid natural cork but I made 2 grips about a year and a half ago out of red burl and I used custom tinted Minwax exterior water based to make them REALLY red and they've held up just fine through moderate use and very little care. I charged the guy about $10 extra to cover the whole cost of the stain and gave him the whole can for touch-ups but it hasn't been needed.

Steve

I had some success tinting cork by taking feather dye mixed with water ,salt and a little vinegar. I put it in a canning jar and boil it and put the lid on it and wait for the lid to pop down. I guess I just canned cork. LOL I let it set for a week rinse off. buff till I get the right patina. and put a wipe on wipe off Urethane coat. It is just a big pain in the butt to do

Good Wraps Bob

Why not just vacuum seal them with the dye. That should draw the stain deep into the cork. Works with other stuff should work here

Rich the cells of cork are so dense that if you use your method. You will still have to put the cork in boiling water to soften it or loosen the tight cell compaction. My wife would kill me if I used our vacuum sealer.So I use our old canner. I am sure your method should work if you soften the cork first. You know it is strange that know one in the cork industry has ever chimed in when cork is being discussed.

Good Wraps Bob

I have colored several with a watery mix of finish with a tint in it. Whip it on. let dry. Then seal

Bill - willierods.com